1. Field of the Invention
The present invention relates generally to differential scanning calorimeters (DSCs), and more specifically to systems and methods for calibrating the contact thermal resistance between pans and sensors in DSCs.
2. Background of the Invention
Differential Scanning Calorimeters measure the heat flow to a sample as the sample temperature is varied in a controlled manner. There are two basic types of DSC, heat flux and power compensation. Brief descriptions of the two types of DSC are included below. A detailed description of the construction and theory of DSCs is disclosed in xe2x80x9cDifferential Scanning Calorimetry an Introduction for Practitionersxe2x80x9d, G. Hxc3x6hne, W. Hemminger and H.-J. Flammersheim (Springer-Verlag, 1996).
Heat flux DSCs include a sensor to measure heat flow to a sample to be analyzed. The sensor has a sample position and a reference position. The sensor is installed in an oven whose temperature is varied dynamically according to a desired temperature program. As the oven is heated or cooled, the temperature difference between the sample and reference positions of the sensor is measured. This temperature difference is assumed to be proportional to the heat flow to the sample.
Power compensation DSCs include a sample and a reference holder installed in a constant temperature enclosure. Each of the holders has a heater and a temperature sensor. The average of the sample and reference holder temperatures is used to control temperature, which follows the desired temperature program. In addition, differential power proportional to the temperature difference between the holders is added to the average power to the sample holder and subtracted from the average power to the reference holder in an effort to reduce the temperature difference between sample and reference holders to zero. The differential power is assumed to be proportional to the sample heat flow and is obtained by measuring the temperature difference between the sample and reference holders. In commercial power compensation DSCs, the difference between sample and reference temperature is generally not zero because a proportional controller is used to control the differential power.
In both heat flux and power compensation DSCs, a sample to be analyzed is loaded into a pan and placed on the sample position of the DSC. An inert reference material may be loaded into a pan and placed on the reference position of the DSC, although usually the reference pan is empty. The temperature program for conventional DSCs typically includes combinations of constant heating (or cooling) rate and constant temperature segments. Modulated DSC (xe2x80x9cMDSCxe2x80x9d), which is described in U.S. Pat. No. 5,224,775 (the xe2x80x9c""775 patentxe2x80x9d), and which is incorporated by reference herein, uses a temperature program in which periodic temperature oscillations are superposed on the constant heating rate and constant temperature segments. The experimental result is the sample heat flow versus temperature or time. The heat flow signal is the result of heat flow to or from the sample due to its specific heat and as a result of transitions occurring in the sample.
For both heat flux and power compensation DSCs, a temperature difference is created between the sample and reference positions of the DSC during the dynamic portion of the DSC experiment. In heat flux DSCs, the temperature difference results principally from the combination of three differential heat flows: the difference between the sample and reference heat flow, the difference between sample and reference sensor heat flow and the difference between sample and reference pan heat flow. In power compensation DSCs, the temperature difference results principally from the combination of three differential heat flows plus the differential power supplied to the sample holders: the difference between the sample and reference heat flow, the difference between sample and reference holder heat flow and the difference between sample and reference pan heat flow. The heat flow difference between the sample and reference consists of heat flow due to the heat capacity difference between the sample and reference, the heat flow of a transition, or the difference in heating rate that occurs during an MDSC experiment. The heat flow difference between the sample and reference sections of the DSC is the result of thermal resistance and capacitance imbalances in the sensor or between the holders, and the difference in heating rate that occurs between the sample and reference sections of the DSC during a transition or during an MDSC experiment. Similarly, the heat flow difference between the sample and reference pans is the result of mass differences between the pans and the difference in heating rate that occurs during a sample transition or during a MDSC experiment.
In conventional heat flux DSCs the sensor imbalance and pan imbalance are assumed to be insignificant and the differences in heating rates are ignored. In conventional power compensation DSCs the holder imbalance and pan imbalance are assumed to be insignificant and the differences in heating rates are ignored. When the balance assumptions are satisfied and the sample heating rate is the same as the reference heating rate, the temperature difference is proportional to the sample heat flow and the differential temperature gives an accurate measure of the sample heat flow. The sample heat flow is only proportional to the measured temperature difference between sample and reference when the heating rate of the sample and reference are identical, the sensor is perfectly symmetrical, and the pan masses are identical. Proportionality of sample heat flow to temperature difference for a balanced sensor and pans occurs only during portions of the experiment when the instrument is operating at a constant heating rate, the sample is changing temperature at the same rate as the instrument and there are no transitions occurring in the sample. During Modulated DSC experiments, the heating rates of the sample and reference are generally not the same and the difference between measured sample and reference temperatures is not proportional to the sample heat flow.
Thus, the sample heat flow from a conventional DSC is not the actual sample heat flow, but includes the effects of imbalances and differences in heating rates; in other words the DSC sample heat flow measurement is smeared. For many DSC experiments, the smeared sample heat flow is a sufficiently accurate result. For example, when the desired experimental result is the total energy of the transition, such as the heat of fusion of a melt, the total peak area is integrated over a suitable baseline and the result from a conventional DSC is sufficiently accurate. If however, partial integration of the peak area is required (for example, in the study of reaction kinetics), the smeared sample heat flow of conventional DSC should not be used. Another example of when the conventional DSC result is inadequate is when two or more transitions in a sample occur within a small temperature interval. In that case, the transitions may be poorly separated in prior art DSCs because of the smearing effects.
During a transition, the heat flow to the sample increases or decreases from the pre-transition value depending upon whether the transition is exothermic or endothermic and whether the DSC is being heated or cooled. The change in sample heat flow causes the heating rate of the sample to be different from that of the DSC and as a consequence, the sample pan and sensor heating rates become different from the programmed heating rate.
U.S. patent applications Ser. Nos. 09/533,949 and 09/643,870, incorporated by reference above, disclose a heat flux DSC that uses a four term heat flow equation to account for sensor imbalances and differences in heating rate between the sample and reference sections of the sensor. The four term DSC heat flow equation derived in the ""949 application is:   q  =            Δ      ⁢              xe2x80x83            ⁢                        T          0                ·                  (                                                    R                r                            -                              R                s                                                                    R                r                            ·                              R                s                                              )                      -                  Δ        ⁢                  xe2x80x83                ⁢        T                    R        r              +                  (                              C            r                    -                      C            s                          )            ·                        ⅆ                      T            s                                    ⅆ          τ                      -                  C        r            ·                                    ⅆ            Δ                    ⁢                      xe2x80x83                    ⁢          T                          ⅆ          τ                    
The first term accounts for the effect of the difference between the sample sensor thermal resistance and the reference sensor thermal resistance. The second term is the conventional DSC heat flow. The third term accounts for the effect of the difference between the sensor sample thermal capacitance and the sensor reference thermal capacitance. The fourth term accounts for the effect of the difference between the heating rates of the sample and reference sides of the DSC.
U.S. patent application Ser. No. 09/643,869, incorporated by reference above, discloses a power compensation DSC that uses a five term heat flow equation to account for sample and reference holder imbalances and differences in heating rate between the sample and reference holders. The five term power compensation DSC heat flow equation derived in the ""869 application is:   q  =            Δ      ⁢              xe2x80x83            ⁢      p        +          Δ      ⁢              xe2x80x83            ⁢                        T          0                ·                  (                                                    R                r                            -                              R                s                                                                    R                r                            ·                              R                s                                              )                      -                  Δ        ⁢                  xe2x80x83                ⁢        T                    R        r              +                  (                              C            r                    -                      C            s                          )            ·                        ⅆ                      T            s                                    ⅆ          τ                      -                  C        r            ·                                    ⅆ            Δ                    ⁢                      xe2x80x83                    ⁢          T                          ⅆ          τ                    
The first term is the difference in power supplied to the sample position versus the power supplied to the reference position. The second term accounts for differences between the thermal resistances of the sample and reference holders. The third term accounts for the heat flow that results from the difference in temperature between the sample and reference holders and is the conventional power compensation DSC heat flow. The fourth term is the heat flow resulting from imbalances in thermal capacitance between the sample and reference holders. The fifth term reflects heat flow resulting from differences in heating rate between the sample and reference holders.
The contact thermal resistances between the sample and reference pans and the DSC sensor are needed to practice the improved methods of DSC and MDSC disclosed in U.S. patent applications Ser. Nos. 09/767,903 and 09/769,313, incorporated by reference above. Two methods for determining the contact thermal resistance are described in these applications. The first method uses the slope of the onset of a first order transition (typically a melt) to find the contact thermal resistance. The second method applies quasi-isothermal MDSC to a sample of known heat capacity. These two methods can be used to obtain typical values of the contact thermal resistances for use in a contact thermal resistance model equation that uses the thermal conductivity of the pans, sensor, DSC purge gas and three geometric parameters to calculate the contact thermal resistance. Using the typical values found by the above two methods, the geometric parameters may be determined and the model equation may be used in heat flow measurements.
However, there is an uncertainty in the heat flow measurement resulting from the use of typical values of contact thermal resistance, because the actual contact thermal resistance during an experiment almost never matches the typical value exactly. Ideally the contact thermal resistance should be determined for each pan and used in the DSC experiment that used that pan. The two methods disclosed in the ""903 and 313 applications may not be used to determine the contact thermal resistance for any given experiment because they require a priori knowledge of sample properties. Either the properties of a transition in the sample in the case of the onset slope method or the sample heat capacity in the MDSC method must be known. Because these properties are the properties that one wishes to determine from DSC experiments, they are not available to determine contact thermal resistance.
The following references may be consulted for additional background information:
A. Boller, Y. Jin and B. Wunderlich, xe2x80x9cHeat Capacity Measurement by Modulated DSC at Constant Temperaturexe2x80x9d, Journal of Thermal Analysis V42 (1994) 307-330.
B. Wunderlich, xe2x80x9cTemperature-Modulated Calorimetry of Polymers with Single and Multiple Frequencies to Determine Heat Capacities as well as Reversible and Irreversible Transition Parametersxe2x80x9d, Material Characterization by Dynamic and Modulated Thermal Analysis Techniques, ASTM STP 1402, A. Riga and L. Judovits, Eds., ASTM, Conshohocken, Pa., 2001.
B. Wunderlich, R. Androsch, M. Pyda and Y. K. Kwon, xe2x80x9cHeat Capacity by Multi-Frequencies Saw-Tooth Modulationxe2x80x9d, Submitted to Thermochimica Acta, September 1999.
R. L. Danley and P. A. Caulfield, xe2x80x9cBaseline Improvements Obtained by a New Heat Flow Measurement Techniquexe2x80x9d, Proceedings 29th N. American Thermal Analysis Society, 2001.
R. L. Danley and P. A. Caulfield, xe2x80x9cDSC Resolution and Dynamic Response Improvements Obtained by a New Heat Flow Measurement Techniquexe2x80x9d, Proceedings 29th N. American Thermal Analysis Society, 2001.
Cp is the heat capacity;
Tp is the sample or reference pan temperature;
Ts is the temperature of the sample sensor;
Tr is the temperature of the reference sensor;
q is the sample or reference heat flow;
q1 is the amplitude of the first heat flow signal;
q2 is the amplitude of the second heat flow signal;
qs and qc are the magnitudes of the sine and cosine components
qs1 is the magnitude of the sine component of the first heat flow signal;
qs2 is the magnitude of the sine component of the second heat flow signal;
qc1 is the magnitude of the cosine component of the first heat flow signal;
qc2 is the magnitude of the cosine component of the second heat flow signal;
Ts and Tc are the magnitudes of the sine and cosine components of the temperature signal, respectively;
Ts1 is the magnitude of the sine component of the first temperature signal;
Tc1 is the magnitude of the cosine component of the first temperature signal;
Ts2 is the magnitude of the sine component of the second temperature signal;
Tc2 is the magnitude of the cosine component of the second temperature signal;
xcfx891 is the angular frequency corresponding to the first period;
xcfx892 is the angular frequency corresponding to the second period;
Rp is the contact thermal resistance;
Rn is the nominal value of the contact thermal resistance;
xcex8 is the product of the angular frequency xcfx89 and the elapsed time t of the experiment; and
Q is the magnitude of the heat flow.
The present invention is a system and method for obtaining the contact thermal resistance for a sample and pan without a priori knowledge of the properties of either the sample or the pan. The method of the present invention involves measuring the reversing heat capacity of a sample and its pan (or an empty pan on the reference side of the DSC) at a long period and a short period during a quasi-isothermal MDSC experiment and finding the value of contact thermal resistance that makes the short and long period heat capacities match. Several different methods may be used to find the contact thermal resistance using quasi-isothermal MDSC with a long and a short period.
The first two methods are direct calculation methods that use the results from an MDSC experiment used with model equations to calculate the contact thermal resistance. Both methods start with the measured, or apparent, heat capacity for a sample and its pan, which is given by:       C    p    =            q      _              ω      ⁢              xe2x80x83            ⁢                        T          _                p            
Where q is the sample or reference heat flow and Tp is the sample or reference pan temperature and the overbar indicates the amplitude of the signal. The pan temperature is found from:
Tp=Txe2x88x92qRp,
where T is the temperature of the sample or reference sensor and Rp is the sample or reference contact thermal resistance. Both the measured temperatures and the measured heat flows are sinusoidal signals with both sine and cosine components:
q=qs sin xcex8+qc cos xcex8
T=Ts sin xcex8+Tc cos xcex8
Coefficients qs and qc are the magnitudes of the sine and cosine components of the heat flow signal. Coefficients Ts and Tc are the magnitudes of the sine and cosine components of the temperature signal, and xcex8 is the product of the angular frequency xcfx89 and the elapsed time t of the experiment. Substituting into the above equation for the pan temperature and collecting sine and cosine components yields:
Tp=(Tsxe2x88x92qsRp)sin xcex8+(Tcxe2x88x92qcRp)cos xcex8
The magnitude of the pan temperature is the square root of the sum of the squares of the sine and cosine components, which is substituted into the heat capacity equation giving:       C    p    =            q      _              ω      ⁢                                                  (                                                T                  s                                -                                                      q                    s                                    ⁢                                      R                    p                                                              )                        2                    +                                    (                                                T                  c                                -                                                      q                    c                                    ⁢                                      R                    p                                                              )                        2                              
The magnitudes of the sine and cosine components of the sensor temperature and the heat flow may be found using the deconvolution technique currently employed in MDSC as disclosed in the ""775 patent or any other technique for separating sine and cosine components of a signal. The first two direct calculation methods diverge at this point. The first method can be used to complete the calculation when the longer of the two periods is so long that the dependence on period essentially vanishes (i.e., vanishes for all practical purposes). In that case, the heat capacity may be found accurately regardless of the value of contact thermal resistance because the sensor and pan temperatures become essentially identical. The value of the heat capacity measured during the long period is substituted for Cp, and the sine and cosine magnitudes of the short period data are used on the right hand side of the equation. The result is a quadratic equation, which may be solved to find Rp.       R    p    =                    -        b            ±                                    b            2                    -                      4            ⁢                          xe2x80x83                        ⁢            ac                                      2      ⁢      a      
Where:
a=qs2+qc2
b=xe2x88x922(Tsqs+Tcqc)
  c  =            T      s      2        +          T      c      2        -          (                                    q            s            2                    +                      q            c            2                                    ω          ⁢                      xe2x80x83                    ⁢                      C            p                              )      
A decision must be made as to which of the two roots is correct. The contact thermal resistance must be real and positive. If both roots are positive, the smaller of two positive roots is the correct root. If one root is negative, then the positive root is the correct root.
The second direct calculation method (which is the preferred method; it is essentially a more general version of the first method) can be used to complete the calculation by using two equations for Cp as above, one each with the sine and cosine components of the temperature and heat flow signals from the long and short period measurements. The heat capacities are eliminated with the result being a quadratic equation in the contact thermal resistance Rp. In that case, the coefficients of the quadratic equation are:       a    =                  q        s2        2            +              q        c2        2            -                                    (                                                            ω                  1                                ⁢                                  q                  2                                                                              ω                  2                                ⁢                                  q                  1                                                      )                    2                ⁢                  (                                    q              s1              2                        +                          q              c1              2                                )                          b    =          -              2        ⁡                  [                                                    q                s2                            ⁢                              T                s2                                      +                                          q                c2                            ⁢                              T                c2                                      -                                                            (                                                                                    ω                        1                                            ⁢                                              q                        2                                                                                                            ω                        2                                            ⁢                                              q                        1                                                                              )                                2                            ⁢                              (                                                                            q                      s1                                        ⁢                                          T                      s1                                                        +                                                            q                      c1                                        ⁢                                          T                      c1                                                                      )                                              ]                                c      =                        T          s2          2                +                  T          c2          2                -                                            (                                                                    ω                    1                                    ⁢                                      q                    2                                                                                        ω                    2                                    ⁢                                      q                    1                                                              )                        2                    ⁢                      (                                          T                s1                2                            +                              T                c1                2                                      )                                ,  
where the subscripts 1 and 2 indicate the results from the long and short period modulations, respectively, and q is the amplitude of the modulated heat flow (not the sine and cosine components).
When the long period part of the calibration experiment is very long, both the first and second methods give the same result. In principle, the second method allows any two periods to be used and does not require one of the periods to be so long that the period dependence vanishes. However, the two periods should be chosen far enough apart so that there will be substantial differences in the magnitudes of the signals. The contact thermal resistances can be tested by running a quasi-isothermal heat capacity experiment at multiple frequencies. If the contact thermal resistance is correct, the sample heat capacity and the sample and reference apparent heat capacities will be essentially independent of modulation period.
A third method is also a direct calculation method. This third method uses the phase angle between the heat flow signal and sensor temperatures, and begins by assuming that the measured heat flow is a cosine function. This assumption is not necessary, but simplifies the calculations. For quasi-isothermal MDSC experiments, when the sample is not reacting or undergoing a transition, the heat flow will be sinusoidal for a sinusoidal temperature modulation:
q=Q cos xcfx89t,
where Q is the magnitude of the heat flow. The heat flowing to the sample and its pan is just the sensible heat associated with the specific heat capacity of the sample and pan:   q  =            C      p        ⁢                  ⅆ                  T          p                            ⅆ        t            
Setting the equations equal gives the sample pan temperature:       T    p    =            Q                        C          p                ⁢        ω              ⁢    sin    ⁢          xe2x80x83        ⁢    ω    ⁢          xe2x80x83        ⁢    t  
As mentioned above, the equation for pan temperature in terms of sensor temperature and the measured heat flow is:
Tp=Txe2x88x92qRp
Substituting the measured heat flow and the pan temperature in terms of the magnitude of the modulated heat flow into the equation for pan temperature and solving for measured temperature yields:   T  =                    QR        p            ⁢      cos      ⁢              xe2x80x83            ⁢      ω      ⁢              xe2x80x83            ⁢      t        +                  Q                  ω          ⁢                      xe2x80x83                    ⁢                      C            p                              ⁢      sin      ⁢              xe2x80x83            ⁢      ω      ⁢              xe2x80x83            ⁢      t      
The tangent of the phase angle between the measured heat flow and measured temperature is the ratio of the magnitude of the sine and cosine components of the measured temperature:       tan    ⁢          xe2x80x83        ⁢    φ    =      1          ω      ⁢              xe2x80x83            ⁢              C        p            ⁢              R        p            
Thus, the pan contact thermal resistance is:       R    p    =      1          ω      ⁢              xe2x80x83            ⁢              C        p            ⁢      tan      ⁢              xe2x80x83            ⁢      φ      
The phase angle between the sensor temperature and the heat flow can be found using the deconvolution method of the original ""775 patent, or any other convenient technique that gives the phase angle. The phase angle is measured during the short period part of the calibration experiment and the heat capacity is found from the long period part of the calibration experiment. This calculation always yields a single positive valued contact thermal resistance.
A fourth and a fifth method for implementing the present invention use curve fitting of the apparent heat capacity for multiple values of pan contact thermal resistance at a short period to determine the contact thermal resistance. The calibration experiments for both of these methods use a long period and a short period quasi-isothermal modulation. In the fourth method, the long period modulation gives values of the sample and reference heat capacities that are assumed to be independent of the pan contact thermal resistance and, hence, independent of period. These values of heat capacity may be considered to be the target values. During the short period part of the calibration experiment, the sensor temperature and the heat flow are separated into their sine and cosine components. Using those signal components, several values (at least four if the function to be fitted is a quadratic, because it has three variables; more than four would be preferable) of contact thermal resistance are used to calculate values of the apparent heat capacity Cxe2x80x2p, using the following equation:       C    p    xe2x80x2    =            q      _              ω      ⁢                                                  (                                                T                  s                                -                                                      q                    s                                    ⁢                                      R                    p                                                              )                        2                    +                                    (                                                T                  c                                -                                                      q                    c                                    ⁢                                      R                    p                                                              )                        2                              
In practice, the number of points used depends upon the fitting method and the function to be fitted. Since, in this method calculates these points from one data set, there is no reason to limit the number of points, except in the sense that increasing the number of points would increase the computational time. In any event, if the number of variables in the function is n, then at least n+1 points are needed, with better results being obtained with a greater number of points. In a preferred embodiment, the resulting data is fitted to a quadratic equation:
Cxe2x80x2p=aRp2+bRp+c
Any suitable routine for curve fitting may be used. For example the xe2x80x9cmethod of least squaresxe2x80x9d is a suitable curve fitting routine. The contact thermal resistance is calculated from the quadratic formula:       R    p    =                    -        b            ±                                    b            2                    -                      4            ⁢                          a              ⁢                              (                                  c                  -                                      C                    p                                                  )                                                                2      ⁢      a      
where Cp is the heat capacity measured at the long period modulation.
This equation gives two positive roots. The smaller root is the correct value. Occasionally, a real solution cannot be found. When that situation occurs, the maximum value of the pan contact thermal resistance from the quadratic fit is used. The maximum value for the quadratic fit is:       R    pmax    =                    -        b                    2        ⁢        a              .  
It has been found that the maximum value typically results in values for the pan contact thermal resistance that yield multiple frequency apparent heat capacities that vary only a few percent, which is a substantial improvement over the methods used in the ""903 or ""313 applications, which use typical values for the pan contact thermal resistance.
The fifth method is similar to the fourth method, but uses two arbitrary (although substantially different) modulation periods. The apparent heat capacity versus contact thermal resistance is fitted for the results from both periods. The heat capacities are eliminated between the two equations, yielding a quadratic equation, which is solved to find the contact thermal resistance. The quadratic equation is:
Rp2(a1xe2x88x92a2)+Rp(b1xe2x88x92b2)+c1xe2x88x92c2=0
Where, a, b and c are the coefficients of the fitted data and the subscripts 1 and 2 indicate the coefficient of each of the two different periods. The contact thermal resistance is given by:       R    p    =                    -                  (                                    b              1                        -                          b              2                                )                    ±                                                  (                                                b                  1                                -                                  b                  2                                            )                        2                    -                      4            ⁢                          (                                                a                  1                                -                                  a                  2                                            )                        ⁢                          (                                                c                  1                                -                                  c                  2                                            )                                                  2      ⁢              (                              a            1                    -                      a            2                          )            
The correct root must be chosen. It is either the positive real root, or the smaller of two real positive roots. In the case where a complex conjugate pair of roots is found, the contact thermal resistance is taken to be the maximum value of the contact thermal resistance equation:       R    p    =            -              (                              b            1                    -                      b            2                          )                    2      ⁢              (                              a            1                    -                      a            2                          )            
In practice, methods one through five may be applied in one of two ways. In the first way, one or more quasi-isothermal calibration experiments are done at one or more temperatures to determine the pan contact thermal resistance at each of the calibration temperatures. Using this data, the DSC or MDSC experiment is performed using these values of contact thermal resistance to measure the heat flow. When multiple temperatures are used, a curve may be fitted to the data, or interpolation may be used between points. This first way is limited to samples that either do not have transitions within the range of the contact thermal resistance calibration or that have fully reversible transitions.
The second way for using the pan contact thermal resistance calibration is to use it to adjust the typical values of pan contact thermal resistance used for a given pan type. In this implementation, the contact thermal resistance calibration is performed at the beginning of a thermal program and the contact thermal resistance obtained is compared to the typical value for that pan type and purge gas at the calibration temperature. The ratio of the calibration value to the typical value is used to scale the typical value during the experiment that follows. Several additional calibration steps may be inserted at desired points within the thermal program.